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anonymous 5 years ago if f(x)=2x-3 and g(x)=2x^2+1 find A.f(g(2)) B. g(f(x)) C. does f(x) have and inverse if so find the inverse of f(x)

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1. anonymous

f surely has an inverse because it is a line and therefore one to one.

2. anonymous

f says 'multiply by 2 then subtract 3' so f inverse would say to do the opposite things in the opposite order: add 3 and divide by 2 so $f^{-1}(x)=\frac{x+3}{2}$

3. anonymous

if this is confusing, rewrite $f(x)=2x-3$ as $y = 2x-3$ then switch x and y (because that is what the inverse does) to get $x=2y-3$ and solve this for x: $x=2y-3$ $x+3=2y$ $\frac{x+3}{2}=y$ and y is your inverse.

4. anonymous

is that the answer for C

5. anonymous

yes. was it clear?

6. anonymous

ok so whats the answers for a and b

7. anonymous

$f(g(x))=f(2x^2+1)=2(2x^2+1)-3$

8. anonymous

i guess there is a little algebra to do now: $2(2x^2+1)-3=4x^2+2-3=4x^2-1$

9. anonymous

but it was f(g(2))

10. anonymous

is it clear what i did? first write $f(g(x))$ then replace $g(x)$ by $2x^2+1$ and then rewrite f replacing $x$ by $2x^2+1$

11. anonymous

oh well if it is $f(g(2))$ then since $f(g(x))=4x^2-1$ then $f(g(2))=4(2^2)+1=17$

12. anonymous

typo sorry. $4(2^2)-1=15$

13. anonymous

or you could say $g(2)=2(2^2)+1=9$ and $f(9)=2\times 9 - 1=15$

14. anonymous

ok

15. anonymous

$g(f(x))=g(2x-3)=2(2x-3)^2+1$

16. anonymous

this requires more algebra: $2(2x-3)^2+1=2(2x-3)(2x-3)+1=2(4x^2-6x+9)+1$ $=8x^2-12x+18+1=8x^2-12x+19$ if my algebra is correct.

17. anonymous

ok for a how come its 4(2^2)-1

18. anonymous

and not 2(2^2) +1

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